Computer Science – Data Structures and Algorithms
Scientific paper
2011-04-09
Computer Science
Data Structures and Algorithms
8 pages
Scientific paper
We prove that for any real-valued matrix $X \in \R^{m \times n}$, and positive integers $r \ge k$, there is a subset of $r$ columns of $X$ such that projecting $X$ onto their span gives a $\sqrt{\frac{r+1}{r-k+1}}$-approximation to best rank-$k$ approximation of $X$ in Frobenius norm. We show that the trade-off we achieve between the number of columns and the approximation ratio is optimal up to lower order terms. Furthermore, there is a deterministic algorithm to find such a subset of columns that runs in $O(r n m^{\omega} \log m)$ arithmetic operations where $\omega$ is the exponent of matrix multiplication. We also give a faster randomized algorithm that runs in $O(r n m^2)$ arithmetic operations.
Guruswami Venkatesan
Sinop Ali Kemal
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